- The Sard theorem for essentially smooth locally Lipschitz maps and applications in optimization Xuan Duc Ha Truong(txdhamath.ac.vn) Abstract: The classical Sard theorem states that the set of critical values of a \$C^{k}\$-map from an open set of \$\R^n\$ to \$\R^p\$ (\$n\geq p\$) has Lebesgue measure zero provided \$k\geq n-p+1\$. In the recent paper by Barbet, Dambrine, Daniilidis and Rifford, the so called ``preparatory Sard theorem" for a compact countable set \$I\$ of \$C^k\$ maps from \$\R^n\$ to \$\R^p\$ and a Sard theorem for a locally Lipschitz continuous selection of this family have been established under the assumption that \$k\geq n-p+1\$. Here, we show that, in the special case \$n=p\$ and \$I\$ is finite, the \$C^1\$ smoothness assumption in these results can be relaxed to ``essentially smooth locally Lipschitz". Then we apply the obtained results to study Karush-Kuhn-Tucker type necessary condition for scalar/vector parametrized constrained optimization problems and the set of Pareto optimal values of a continuous selection of a finite family of essentially smooth locally Lipschitz maps. Keywords: Sard theorem, locally Lipschitz, essentially smooth map, critical points, optimization Category 1: Convex and Nonsmooth Optimization Category 2: Other Topics (Multi-Criteria Optimization ) Citation: Download: [PDF]Entry Submitted: 12/04/2018Entry Accepted: 12/05/2018Entry Last Modified: 12/04/2018Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.